SOLUTION: . If x = −2 is a zero of f (x) = x3 + 6x2 + 11x + 6, then f (x) factors
completely as ( use synthetic or long division; show your work) (LO 12)
a) (x − 2)(x + 3)(x +
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Question 117393: . If x = −2 is a zero of f (x) = x3 + 6x2 + 11x + 6, then f (x) factors
completely as ( use synthetic or long division; show your work) (LO 12)
a) (x − 2)(x + 3)(x + 1);
b) (x + 2)(x + 3)(x − 1);
c) (x − 2)(x + 3)(x − 1);
d) (x + 2)(x + 3)(x + 1);
I am lost with the synthetic division
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Let's set up the synthetic division table by placing the test zero in the upper left corner and placing the coefficients of to the right of the test zero.
Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 1)
Multiply -2 by 1 and place the product (which is -2) right underneath the second coefficient (which is 6)
Add -2 and 6 to get 4. Place the sum right underneath -2.
Multiply -2 by 4 and place the product (which is -8) right underneath the third coefficient (which is 11)
Add -8 and 11 to get 3. Place the sum right underneath -8.
Multiply -2 by 3 and place the product (which is -6) right underneath the fourth coefficient (which is 6)
Add -6 and 6 to get 0. Place the sum right underneath -6.
Since the last column adds to zero, we have a remainder of zero. This means is a factor of
Now lets look at the bottom row of coefficients:
The first 3 coefficients (1,4,3) form the quotient
So
You can use this online polynomial division calculator to check your work
Basically factors to
Now lets break down further
Looking at we can see that the first term is and the last term is where the coefficients are 1 and 3 respectively.
Now multiply the first coefficient 1 and the last coefficient 3 to get 3. Now what two numbers multiply to 3 and add to the middle coefficient 4? Let's list all of the factors of 3:
Factors of 3:
1,3
-1,-3 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to 3
1*3
(-1)*(-3)
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to 4? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 4
| First Number | Second Number | Sum | | 1 | 3 | 1+3=4 |
| -1 | -3 | -1+(-3)=-4 |
From this list we can see that 1 and 3 add up to 4 and multiply to 3
Now looking at the expression , replace with (notice adds up to . So it is equivalent to )
Now let's factor by grouping:
Group like terms
Factor out the GCF of out of the first group. Factor out the GCF of out of the second group
Since we have a common term of , we can combine like terms
So factors to
So this also means that factors to (since is equivalent to )
So factors to
Now reintroduce the first factor
-------------------------------
Answer:
So factors to which means the answer is D)
Notice if we graph we get
Graph of
and if we graph , we get
Graph of
and you can see that the two graphs would overlap each other if they were plotted on the same screen. So this means that the two polynomials are equivalent. So this visually verifies our answer.
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